Combining Short Multiplication Formulas - Examples, Exercises and Solutions

Understanding Combining Short Multiplication Formulas

Complete explanation with examples

Combining the square of a sum formulas

Meet the square of binomials formulas:

Product of the sum of two terms and their difference

(X+Y)(XY)=X2Y2(X + Y)\cdot(X - Y) = X^2 - Y^2

Click here to read more about Multiplication of the sum of two elements by the difference between them!

The difference of squares formula

(XY)2=X22XY+Y2(X - Y)^2=X^2 - 2XY + Y^2

Click here to read more about the formula for the difference of squares

The Square of a Sum Formula

(X+Y)2=X2+2XY+Y2(X + Y)^2=X^2+ 2XY + Y^2

Click here to read more about the formula for the Sum of Squares

Formulas relating to two expressions cubed

(a+b)3=a3+3a2b+3ab2+b3(a+b)^3=a^3+3a^2 b+3ab^2+b^3

(ab)3=a33a2b+3ab2b3(a-b)^3=a^3-3a^2 b+3ab^2-b^3

Click here to read more about Formulas for Cubic Expressions

Example

Practice an exercise that combines all the shortened multiplication formulas together:
x2+(5+x)(5x)+x26x+9(63+362x+36x2+x3)=(2+x)2+(6+x)3x^2+(5+x)(5-x)+x^2-6x+9-(6^3+3\cdot 6^2\cdot x+3\cdot 6\cdot x^2+x^3)=(2+x)^2+(6+x)^3

Let's start from the beginning of the exercise. Observe that the expression
(5+x)(5x)=25x2(5+x)(5-x)=25-x^2
matches the product of the sum of th two terms and their difference
(X+Y)(XY)=X2Y2(X + Y)\cdot(X - Y) = X^2 - Y^2
Let's proceed to the expression x26x+9x^2-6x+9
We notice that it matches the difference of squares formula
(XY)2=X22XY+Y2(X - Y)^2=X^2 - 2XY + Y^2
However let's avoid touching it at this stage.
Let's continue to 63+362x+36x2+x36^3+3\cdot 6^2\cdot x+3\cdot 6\cdot x^2+x^3
and we can see that it perfectly matches the formula for two terms cubed
(a+b)3=a3+3a2b+3ab2+b3(a+b)^3=a^3+3a^2 b+3ab^2+b^3
which means
63+362x+36x2+x3=(6+x)36^3+3\cdot 6^2\cdot x+3\cdot 6\cdot x^2+x^3=(6+x)^3

Let's continue to the second side and observe that the expression (2+x)2(2+x)^2 matches the formula for sum of squares
(X+Y)2=X2+2XY+Y2(X + Y)^2=X^2+ 2XY + Y^2
Therefore
(2+x)2=4+4x+x2(2+x)^2=4+4x+x^2

Now let's insert the data:
x2+25x2+x26x+9(6+x)3=4+4x+x2+(6+x)3x^2+25-x^2+x^2-6x+9-(6+x)^3=4+4x+x^2+(6+x)^3
Let's reduce the terms and solve as follows:
6x+34=4x+4-6x+34=4x+4
Move the terms to opposite sides:
8x=308x=30
x=3.75x=3.75

Detailed explanation

Practice Combining Short Multiplication Formulas

Test your knowledge with 1 quizzes

Solve the equation

\( 2x^2-2x=(x+1)^2 \)

Examples with solutions for Combining Short Multiplication Formulas

Step-by-step solutions included
Exercise #1

Solve the following equation:

(x4)2+3x2=16x+12 (x-4)^2+3x^2=-16x+12

Step-by-Step Solution

To solve the given equation, follow these steps:

  • Step 1: Expand (x4)2(x - 4)^2 using the formula (ab)2=a22ab+b2(a - b)^2 = a^2 - 2ab + b^2.

Thus, (x4)2=x28x+16(x - 4)^2 = x^2 - 8x + 16.

  • Step 2: Substitute the expanded form into the equation:

x28x+16+3x2=16x+12x^2 - 8x + 16 + 3x^2 = -16x + 12.

  • Step 3: Combine like terms on the left-hand side.

This gives 4x28x+16=16x+124x^2 - 8x + 16 = -16x + 12.

  • Step 4: Rearrange the equation to set it to zero.

Bring all terms to one side: 4x28x+16+16x12=04x^2 - 8x + 16 + 16x - 12 = 0.

Combine and simplify the terms: 4x2+8x+4=04x^2 + 8x + 4 = 0.

  • Step 5: Simplify the equation by dividing each term by 4.

It becomes x2+2x+1=0x^2 + 2x + 1 = 0.

  • Step 6: Recognize the equation as a perfect square trinomial.

(x+1)2=0(x + 1)^2 = 0.

  • Step 7: Solve by taking the square root of both sides.

The solution is x+1=0x + 1 = 0, therefore x=1x = -1.

In conclusion, the solution to the equation is x=1 x = -1 .

Answer:

x=1 x=-1

Video Solution
Exercise #2

Solve the following equation:

(x+2)2=(2x+3)2 (x+2)^2=(2x+3)^2

Step-by-Step Solution

We will solve the equation (x+2)2=(2x+3)2 (x+2)^2 = (2x+3)^2 by expanding and simplifying both sides:

Step 1: Expand both sides of the equation:
Left side: (x+2)2=x2+4x+4 (x+2)^2 = x^2 + 4x + 4
Right side: (2x+3)2=4x2+12x+9 (2x+3)^2 = 4x^2 + 12x + 9

Step 2: Set the expanded forms equal to each other:
x2+4x+4=4x2+12x+9 x^2 + 4x + 4 = 4x^2 + 12x + 9

Step 3: Rearrange to form a standard quadratic equation:
Subtract x2+4x+4 x^2 + 4x + 4 from both sides:
0=3x2+8x+5 0 = 3x^2 + 8x + 5

Step 4: Rearrange to get:
3x2+8x+5=0 3x^2 + 8x + 5 = 0

Step 5: Solve using the quadratic formula:
Using a=3 a = 3 , b=8 b = 8 , c=5 c = 5 :
x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
x=8±8243523 x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}
x=8±64606 x = \frac{-8 \pm \sqrt{64 - 60}}{6}
x=8±46 x = \frac{-8 \pm \sqrt{4}}{6}
x=8±26 x = \frac{-8 \pm 2}{6}

Step 6: Calculate the solutions:
x1=8+26=66=1 x_1 = \frac{-8 + 2}{6} = \frac{-6}{6} = -1
x2=826=106=53 x_2 = \frac{-8 - 2}{6} = \frac{-10}{6} = -\frac{5}{3}

Verify in the original equation to assure correctness. Hence, both solutions are valid.

Therefore, the solutions are x1=1 x_1 = -1 and x2=53 x_2 = -\frac{5}{3} , which matches choice 3.

Answer:

x1=1,x2=53 x_1=-1,x_2=-\frac{5}{3}

Video Solution
Exercise #3

Solve the equation

2x22x=(x+1)2 2x^2-2x=(x+1)^2

Step-by-Step Solution

The given equation is:

2x22x=(x+1)2 2x^2 - 2x = (x+1)^2

Step 1: Expand the right-hand side.

(x+1)2=x2+2x+1(x+1)^2 = x^2 + 2x + 1

Step 2: Write the full equation with the expanded form.

2x22x=x2+2x+12x^2 - 2x = x^2 + 2x + 1

Step 3: Bring all terms to one side of the equation to set it to zero.

2x22xx22x1=02x^2 - 2x - x^2 - 2x - 1 = 0

Step 4: Simplify the equation.

x24x1=0x^2 - 4x - 1 = 0

Step 5: Identify coefficients for the quadratic formula.

Here, a=1a = 1, b=4b = -4, c=1c = -1.

Step 6: Apply the quadratic formula.

x=(4)±(4)241(1)21x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}

x=4±16+42x = \frac{4 \pm \sqrt{16 + 4}}{2}

x=4±202x = \frac{4 \pm \sqrt{20}}{2}

x=4±252x = \frac{4 \pm 2\sqrt{5}}{2}

x=2±5x = 2 \pm \sqrt{5}

Therefore, the solutions are x=2+5x = 2 + \sqrt{5} and x=25x = 2 - \sqrt{5}.

These solutions correspond to choice (4): Answers a + b

Answer:

Answers a + b

Video Solution
Exercise #4

Find X

7x+1+(2x+3)2=(4x+2)2 7x+1+(2x+3)^2=(4x+2)^2

Step-by-Step Solution

To solve the equation 7x+1+(2x+3)2=(4x+2)2 7x + 1 + (2x + 3)^2 = (4x + 2)^2 , we follow these steps:

  • Step 1: Expand both sides using the square of a binomial formula.
  • Step 2: Simplify the equation to form a standard quadratic equation.
  • Step 3: Use the quadratic formula to find the roots of the equation.

Step 1: Expand the squares.

The left side: (2x+3)2=4x2+12x+9 (2x + 3)^2 = 4x^2 + 12x + 9 .

The right side: (4x+2)2=16x2+16x+4 (4x + 2)^2 = 16x^2 + 16x + 4 .

Step 2: Substitute back into the original equation and simplify:

7x+1+4x2+12x+9=16x2+16x+4 7x + 1 + 4x^2 + 12x + 9 = 16x^2 + 16x + 4 .

Combine like terms:

4x2+19x+10=16x2+16x+4 4x^2 + 19x + 10 = 16x^2 + 16x + 4 .

Step 3: Move all terms to one side:

4x2+19x+1016x216x4=0 4x^2 + 19x + 10 - 16x^2 - 16x - 4 = 0 .

Which simplifies to:

12x2+3x+6=0-12x^2 + 3x + 6 = 0 .

Step 4: Divide by -3 to simplify:

4x2x2=0 4x^2 - x - 2 = 0 .

Step 5: Use the quadratic formula:

x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} , where a=4 a = 4 , b=1 b = -1 , c=2 c = -2 .

Calculate the discriminant:

b24ac=(1)244(2)=1+32=33 b^2 - 4ac = (-1)^2 - 4 \cdot 4 \cdot (-2) = 1 + 32 = 33 .

Calculate the roots:

x=1±338 x = \frac{1 \pm \sqrt{33}}{8} .

Therefore, the solution to the problem is x=1±338 x = \frac{1 \pm \sqrt{33}}{8} .

Answer:

1±338 \frac{1\pm\sqrt{33}}{8}

Video Solution
Exercise #5

Solve the following equation:

(x+3)2=2x+5 (x+3)^2=2x+5

Step-by-Step Solution

To solve the equation (x+3)2=2x+5 (x+3)^2 = 2x + 5 , we proceed as follows:

  • Step 1: Expand the left side. Using the identity (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 , we find:
    (x+3)2=x2+6x+9 (x+3)^2 = x^2 + 6x + 9 .

  • Step 2: Set the equation to zero by moving all terms to one side:
    x2+6x+9=2x+5 x^2 + 6x + 9 = 2x + 5
    Subtract 2x+5 2x + 5 from both sides:
    x2+6x+92x5=0 x^2 + 6x + 9 - 2x - 5 = 0
    This simplifies to:
    x2+4x+4=0 x^2 + 4x + 4 = 0 .

  • Step 3: Solve the quadratic equation x2+4x+4=0 x^2 + 4x + 4 = 0 . Notice this can be factored as:
    (x+2)2=0 (x+2)^2 = 0 .

  • Step 4: Solve for x x by setting the factor equal to zero:
    x+2=0 x+2 = 0 .
    Thus, x=2 x = -2 .

Therefore, the solution to the equation (x+3)2=2x+5 (x+3)^2 = 2x + 5 is x=2 x = -2 .

Answer:

x=2 x=-2

Video Solution

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